A note on radial solutions to the critical Lane-Emden equation with a variable coefficient

In this note, we consider the following problem, \begin{equation*} \begin{cases} -\Delta u=(1+g(x))u^{\frac{N+2}{N-2}},\ u>0\text{ in }B,\\ u=0\text{ on }\partial B, \end{cases} \end{equation*} where $N\ge3$ and $B\subset \mathbb{R}^N$ is a unit ball centered at the origin and $g(x)$ is a radial H\"{o}lder continuous function such that $g(0)=0$. We prove the existence and nonexistence of radial solutions by the variational method with the concentration compactness analysis and the Pohozaev identity.